340 AAE Dynamics and Vibrations
Problem Set 1
Due: 9/18/2012
On the 340 Blackboard site, a review of dimensions is included in a document: Supplementary
Material 203 Review Material Dimensional Analysis. After reading the document,
consider the following problems:
Problem 1: Determine the appropriate dimensions in these examples.
(a) A particle moves under the influence of two forces such that its location is determined from
the following relationship
y m s ln 1 s
k
Given dim y L, dim m M , and dim s L , determine the dimensions of , k , .
T
(b) The following equation describes motions of a spacecraft in orbit
G
2
dx
u 2Gq qH 2 1 3sin Gq c 0 where G 2 dy
dt
D
D
y0
Assume
1
F
dim H
T
L
dim q L dim D L
M
dim
dim c F
L
dim u
Determine the dimensions of , , , y and x .
y1
Problem 2: A mechanism is comprised of two rigid bars OS (length h) and SU (length ) that
move in the same plane and are pinned at O and S. Unit vectors are defined such that ei are
inertially-fixed, a are fixed in bar OS b and are fixed in bar SU.
k
j
b1
e3
S
e2
b3
a1
a2
U
O
(a) Derive the unit vector relationships, that is, write unit vectors a1 , a2 , a3 in terms of both
e , e , e and b , b , b .
1
2
3
1
2
3
Write b1 , b2 , b3 in terms of e1 , e2 , e3 .
Express the angular velocities e a , a b .
(b) The motion of interest is the path of S. Thus, point S is the terminal point of a position vector.
But consider two different base points: O and U.
Write position vectors r OS and r US and express each in terms of a1 , a2 , a3 .
The motion of point S can be viewed by different observers as well. Write expressions for the
following quantities: ev OS , a v OS , b v OS , e v US , a v US , b v US . Express each velocity in terms of
a1 , a2 , a3 .
Which velocities are generic? How do you know?
Ans : e v OS h a2
e US
v b1 sin a1 cos a2
(c) Let h 1 met, = 0.5 met. At a certain instant, 45 , .5rad / s, = 30 , .2rad / s .
Evaluate the velocities in (b) and determine the speed for each. Are they the same?